Vacuum degeneracy of chiral spin states in compactified space

A chiral spin state is not only characterized by the T and P order parameter $E_{123}$${=\mathrm{S}}_1$⋅${(\mathrm{S}}_2$×${\mathrm{S}}_3$), it is also characterized by an integer k. In this paper we show that this integer k can be determined from the vacuum degeneracy of the chiral spin state on compactified spaces. On a Riemann surface with genus g the vacuum degeneracy of the chiral spin state is found to be 2$k^g$. Among those vacuum states, some $k^g$ states have 〈$E_{123}$〉>0, while other $k^g$ states have 〈$E_{123}$〉<0. The dependence of the vacuum degeneracy on the topology of the space reflects some sort of topological ordering in the chiral spin state. In general the topological ordering in a system is classified by topological theories.